Theorem nmoofval 29124

Description: The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nmoofval.1	⊢ 𝑋 = (BaseSet‘𝑈)
nmoofval.2	⊢ 𝑌 = (BaseSet‘𝑊)
nmoofval.3	⊢ 𝐿 = (normCV‘𝑈)
nmoofval.4	⊢ 𝑀 = (normCV‘𝑊)
nmoofval.6	⊢ 𝑁 = (𝑈 normOpOLD 𝑊)

Assertion

Ref	Expression
nmoofval	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )))

Distinct variable groups:   𝑥,𝑡,𝑧,𝑈   𝑡,𝑊,𝑥,𝑧   𝑡,𝑋,𝑧   𝑡,𝑌,𝑥   𝑡,𝐿   𝑡,𝑀

Allowed substitution hints:   𝐿(𝑥,𝑧)   𝑀(𝑥,𝑧)   𝑁(𝑥,𝑧,𝑡)   𝑋(𝑥)   𝑌(𝑧)

Proof of Theorem nmoofval

Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmoofval.6	. 2 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)
2		fveq2 6774	. . . . . 6 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3		nmoofval.1	. . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈)
4	2, 3	eqtr4di 2796	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5	4	oveq2d 7291	. . . 4 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑m 𝑋))
6		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
7		nmoofval.3	. . . . . . . . . . 11 ⊢ 𝐿 = (normCV‘𝑈)
8	6, 7	eqtr4di 2796	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = 𝐿)
9	8	fveq1d 6776	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢)‘𝑧) = (𝐿‘𝑧))
10	9	breq1d 5084	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (((normCV‘𝑢)‘𝑧) ≤ 1 ↔ (𝐿‘𝑧) ≤ 1))
11	10	anbi1d 630	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧))) ↔ ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))))
12	4, 11	rexeqbidv 3337	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧))) ↔ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))))
13	12	abbidv 2807	. . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))} = {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))})
14	13	supeq1d 9205	. . . 4 ⊢ (𝑢 = 𝑈 → sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < ))
15	5, 14	mpteq12dv 5165	. . 3 ⊢ (𝑢 = 𝑈 → (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )) = (𝑡 ∈ ((BaseSet‘𝑤) ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )))
16		fveq2 6774	. . . . . 6 ⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
17		nmoofval.2	. . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊)
18	16, 17	eqtr4di 2796	. . . . 5 ⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
19	18	oveq1d 7290	. . . 4 ⊢ (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑m 𝑋) = (𝑌 ↑m 𝑋))
20		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = (normCV‘𝑊))
21		nmoofval.4	. . . . . . . . . . 11 ⊢ 𝑀 = (normCV‘𝑊)
22	20, 21	eqtr4di 2796	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = 𝑀)
23	22	fveq1d 6776	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((normCV‘𝑤)‘(𝑡‘𝑧)) = (𝑀‘(𝑡‘𝑧)))
24	23	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)) ↔ 𝑥 = (𝑀‘(𝑡‘𝑧))))
25	24	anbi2d 629	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧))) ↔ ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))))
26	25	rexbidv 3226	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧))) ↔ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))))
27	26	abbidv 2807	. . . . 5 ⊢ (𝑤 = 𝑊 → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))} = {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))})
28	27	supeq1d 9205	. . . 4 ⊢ (𝑤 = 𝑊 → sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < ))
29	19, 28	mpteq12dv 5165	. . 3 ⊢ (𝑤 = 𝑊 → (𝑡 ∈ ((BaseSet‘𝑤) ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )) = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )))
30		df-nmoo 29107	. . 3 ⊢ normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )))
31		ovex 7308	. . . 4 ⊢ (𝑌 ↑m 𝑋) ∈ V
32	31	mptex 7099	. . 3 ⊢ (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )) ∈ V
33	15, 29, 30, 32	ovmpo 7433	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 normOpOLD 𝑊) = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )))
34	1, 33	eqtrid 2790	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065   class class class wbr 5074   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  supcsup 9199  1c1 10872  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010  NrmCVeccnv 28946  BaseSetcba 28948  norm_CVcnmcv 28952   normOp_OLD cnmoo 29103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-sup 9201  df-nmoo 29107

This theorem is referenced by:  nmooval  29125  hhnmoi  30263